浸入
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數學上,浸入是微分流形之間的可微映射,其導數處處是單射。確切而言,f : M → N是浸入,若在M中每一點p,
都是单射。(TpX表示X在點p處的切空間。另一個等價說法是f是浸入,若f的秩是常數,且等於M的維數:
以上只要求f的導數為單射,但映射f未必是單射。
一個與浸入相關的概念是嵌入。光滑嵌入是一個單射浸入f : M → N而同時為拓撲嵌入,使得M與其在N中的像微分同胚。浸入正是局部嵌入,即對M中每一點x都有一個x的鄰域U ⊂ M,使得f : U → N是嵌入。相反地,局部嵌入都是浸入。
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若M是緊緻的,則單射浸入是一個嵌入;若M不是緊緻,則未必成立。這兩者的關係就如同連續雙射之於同胚。
參考
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